What's the difference between oxidation and reduction? My teacher said something about gaining/losing oxygen or hydrogen, but then also electrons, and I'm lost.

Oxidation and reduction are complementary processes that always occur simultaneously in what's called a redox reaction. You can understand them through different definitions:

Oxidation is:

1. The gain of oxygen by a substance.
2. The loss of hydrogen by a substance.
3. The loss of electrons by a substance (remember LEO: Loss of Electrons is Oxidation).

Reduction is:

1. The loss of oxygen by a substance.
2. The gain of hydrogen by a substance.
3. The gain of electrons by a substance (remember GER: Gain of Electrons is Reduction).

The electron transfer definition is the most fundamental and universally applicable. For example, when magnesium burns in oxygen ($2Mg + O_2 \rightarrow 2MgO$), magnesium is oxidized because it gains oxygen and loses electrons, while oxygen is reduced because it gains electrons from magnesium. They are two sides of the same chemical coin.